If you book a tour through a travel agency back home, then you shouldn't have to worry at all about visas and registration. If, however, you're traveling independently, then you have a number of more or less complicated and expensive options to choose from in order to get a valid visa for your trip. There are three basic types of visa - Tourist, Business and Private - and the one you choose will be dictated less by the purpose of your trip and more by the length of your stay, your connections in Russia and the amount of time and money you have to get the visa. A number of private companies offer visa support for Russia, which can save you the trouble of organizing an invitation, queuing at your local consulate and dealing with often surly consular officials. These services do not normally come cheap, however.
There is a standard application form for all types of visa, which you can download below. Whatever type of visa you choose, you will require an official invitation or tourist voucher and you will still need to obtain and complete a Migration Card at your point of arrival. You will normally be registered either by your hotel, by your tour agent, or by your landlord if you are renting a flat. If you have to do this yourself, you must register within three working days - and, if you're traveling elsewhere in Russia, in any other city where you spend more than three working days.
To apply for a Russian visa you must have a passport valid for at least six months after the proposed date of your departure from Russia, and the passport must have one blank page (or two or three, depending on your nationality).
A postal application can be made for all types of visa, saving you the hassle of going to a consulate in person. It does, however, require extra time, and particular attention to detail.
Making photocopies of all the documents you submit to and obtain from the Russian authorities is strongly recommended.